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This paper introduces a new modeling framework for the statistical analysisof point patterns on a manifold M_{d}, defined by a connected and compacttwo-point homogeneous space, including the special case of the sphere. Thepresented approach is based on temporal Cox processes driven by aL^{2}(\mathbb{M}_{d})-valued log-intensity. Different aggregation schemes onthe manifold of the spatiotemporal point-referenced data are implemented interms of the time-varying discrete Jacobi polynomial transform of the log-riskprocess. The n-dimensional microscale point pattern evolution in time atdifferent manifold spatial scales is then characterized from such a transform.The simulation study undertaken illustrates the construction of spherical pointprocess models displaying aggregation at low Legendre polynomial transformfrequencies (large scale), while regularity is observed at high frequencies(small scale). K-function analysis supports these results under temporalshort-, intermediate- and long-range dependence of the log-risk process.

Point pattern analysis and classification on compact two-point homogeneous spaces evolving time